[This is a quick post I wrote while my younger son was at a little math enrichment activity. Sorry it looks like it was written in a hurry and not proof read . . . ]
Earlier today my older son and I played around with Patrick Honner’s Pi Day exercise:
That project is here:
After we finished my son wondered about extending the exercise to 4 dimensions!
But, extending to 4 dimensions isn’t as easy as it seems. For one thing, the “volume” and “surface area” of a 4 dimensional sphere involve $\latex \pi^2$ not :
“Surface Area” =
So, we’ll modify Honner’s 3d formula to be = (1/128) (Surface Area^4) / (Volume^3). That’ll give us a value for and then we can compute .
So, I found the “volume” and “surface area” of the 4 dimensional regular Polytopes here:
Calculating for the regular 4 dimensional polytopes gave values of approximately:
We’ve actually made a 3D version of the 120-cell with our Zometool set:
That project is here, and maybe helps see that the shape is getting sort of spherical.
Another way to see some of these 4-dimensional shapes is to check out the game Hypernom:
Anyway, thanks for Patrick Honner for a fun Pi day!